11
The profits of the two firms and the welfare of the group that patronizes the co-op are then:
(24)
'
ΠI(3) =
(t-cI+cC
2
pC+pI)
8t
(25)
ΠC(3)
=0
(26) MW('3) =(pC
-cC
cf )xC(3)
—tx.
'2
C(3)
Innovation Competition in the Mixed Oligopsony
In this stage, the two firms seek to determine their optimal innovation effort. Similar to the pure
oligopsony case, the problem of the IOF is to determine the amount of innovation that maximizes its post-
innovation profits minus its innovation costs, i.e.,
(27)
max ΠΙ(2,3) = ΠI(3) -I
(t + rI - rC - Pc + Pi )2
81
1 2
- 2ψ ψrι
On the other hand, the problem of the co-op is to maximize the welfare of farmers that are
members at the time the decision to invest in innovation is being made (this group will be referred to as
the “pre-innovation membership”). As will be shown below (in stage 1), the pre-innovation membership
is the group that, by selling to the co-op at reduced prices in the pre-innovation stage, provides the co-op
with earnings that finance its subsequent cost-reducing innovation effort. Thus, even though the co-op
knows that its cost-reducing process innovation activity will result in increased input pricing that can
attract new farmers/members to the co-op at the post-innovation stage, when making its innovation
decisions the co-op considers only the welfare of farmers that finance its innovation activity (by
patronizing the co-op in stage 1).
Algebraically, the problem of the coop can be expressed as:
where xC(1) is the share of the co-op in stage 1, MW(2,3) is the welfare of the pre-innovation membership
in stages 2 and 3, and MW('3/1) is the welfare of the pre-innovation membership in stage 3. Solving the
problems of the co-op and IOF, we get their best response functions as:
(28)
max MW(2,3)
=MW('3/1)-IC=(PC
-c+rC
-cf ) xC(1)
1 ,2 1 2
2 txC (1) 2 ψrc
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